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

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

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:

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

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

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

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

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

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

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

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

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

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)

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

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

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

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:

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

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:

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)

Sample selection in stochastic frontier estimation using cross-section data

Description

sfaselectioncross is a symbolic formula based function for the estimation of the stochastic frontier model in the presence of sample selection. The model accommodates cross-sectional or pooled cross-sectional data. The model can be estimated using different quadrature approaches or maximum simulated likelihood (MSL). See Greene (2010).

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).

Usage

sfaselectioncross(
  selectionF,
  frontierF,
  uhet,
  vhet,
  modelType = "greene10",
  logDepVar = TRUE,
  data,
  subset,
  weights,
  wscale = TRUE,
  S = 1L,
  udist = "hnormal",
  start = NULL,
  method = "bfgs",
  hessianType = 2L,
  lType = "ghermite",
  Nsub = 100,
  uBound = Inf,
  simType = "halton",
  Nsim = 100,
  prime = 2L,
  burn = 10,
  antithetics = FALSE,
  seed = 12345,
  itermax = 2000,
  printInfo = FALSE,
  intol = 1e-06,
  tol = 1e-12,
  gradtol = 1e-06,
  stepmax = 0.1,
  qac = "marquardt"
)

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

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

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

Arguments

Details

The current model is an extension of Heckman (1976, 1979) sample selection model to nonlinear models particularly stochastic frontier model. The model has first been discussed in Greene (2010), and an application can be found in Dakpo et al. (2021). Practically, we have:

y_{1i} = \left\{ \begin{array}{ll} 1 & \mbox{if} \quad y_{1i}^* > 0 \\ 0 & \mbox{if} \quad y_{1i}^* \leq 0 \\ \end{array} \right.

where

y_{1i}^*=\mathbf{Z}_{si}^{\prime} \mathbf{\gamma} + w_i, \quad w_i \sim \mathcal{N}(0, 1)

and

y_{2i} = \left\{ \begin{array}{ll} y_{2i}^* & \mbox{if} \quad y_{1i}^* > 0 \\ NA & \mbox{if} \quad y_{1i}^* \leq 0 \\ \end{array} \right.

where

y_{2i}^*=\mathbf{x_{i}^{\prime}} \mathbf{\beta} + v_i - Su_i, \quad v_i = \sigma_vV_i \quad \wedge \quad V_i \sim \mathcal{N}(0, 1), \quad u_i = \sigma_u|U_i| \quad \wedge \quad U_i \sim \mathcal{N}(0, 1)

y_{1i} describes the selection equation while y_{2i} represents the frontier equation. The selection bias arises from the correlation between the two symmetric random components v_i and w_i:

(v_i, w_i) \sim \mathcal{N}_2\left\lbrack(0,0), (1, \rho \sigma_v, \sigma_v^2) \right\rbrack

Conditionaly on |U_i|, the probability associated to each observation is:

Pr \left\lbrack y_{1i}^* \leq 0 \right\rbrack^{1-y_{1i}} \cdot \left\lbrace f(y_{2i}|y_{1i}^* > 0) \times Pr\left\lbrack y_{1i}^* > 0 \right\rbrack \right\rbrace^{y_{1i}}

Using the conditional probability formula:

P\left(A\cap B\right) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)

Therefore:

f(y_{2i}|y_{1i}^* \geq 0) \cdot Pr\left\lbrack y_{1i}^* \geq 0\right\rbrack = f(y_{2i}) \cdot Pr(y_{1i}^* \geq 0|y_{2i})

Using the properties of a bivariate normal distribution, we have:

y_{i1}^* | y_{i2} \sim N\left(\mathbf{Z_{si}^{\prime}} \bm{\gamma}+\frac{\rho}{ \sigma_v}v_i, 1-\rho^2\right)

Hence conditionally on |U_i|, we have:

f(y_{2i}|y_{1i}^* \geq 0) \cdot Pr\left\lbrack y_{1i}^* \geq 0\right\rbrack = \frac{1}{\sigma_v}\phi\left(\frac{v_i}{\sigma_v}\right)\Phi\left(\frac{ \mathbf{Z_{si}^{\prime}} \bm{\gamma}+\frac{\rho}{\sigma_v}v_i}{ \sqrt{1-\rho^2}}\right)

The conditional likelihood is equal to:

L_i\big||U_i| = \Phi(-\mathbf{Z_{si}^{\prime}} \bm{\gamma})^{1-y_{1i}} \times \left\lbrace \frac{1}{\sigma_v}\phi\left(\frac{y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|}{\sigma_v}\right)\Phi\left(\frac{ \mathbf{Z_{si}^{\prime}} \bm{\gamma}+\frac{\rho}{\sigma_v}\left(y_{2i}- \mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|\right)}{\sqrt{1-\rho^2}} \right) \right\rbrace ^{y_{1i}}

Since the non-selected observations bring no additional information, the conditional likelihood to be considered is:

L_i\big||U_i| = \frac{1}{\sigma_v}\phi\left(\frac{y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|}{\sigma_v}\right) \Phi\left(\frac{\mathbf{Z_{si}^{\prime}} \bm{\gamma}+\frac{\rho}{\sigma_v}\left(y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|\right)}{\sqrt{1-\rho^2}}\right)

The unconditional likelihood is obtained by integrating |U_i| out of the conditional likelihood. Thus

L_i\\ = \int_{|U_i|} \frac{1}{\sigma_v}\phi\left(\frac{y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|}{\sigma_v}\right) \Phi\left(\frac{\mathbf{Z_{si}^{\prime}} \bm{\gamma}+ \frac{\rho}{\sigma_v}\left(y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|\right)}{\sqrt{1-\rho^2}}\right)p\left(|U_i|\right)d|U_i|

To simplifiy the estimation, the likelihood can be estimated using a two-step approach. In the first step, the probit model can be run and estimate of \gamma can be obtained. Then, in the second step, the following model is estimated:

L_i\\ = \int_{|U_i|} \frac{1}{\sigma_v}\phi\left(\frac{y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|}{\sigma_v}\right) \Phi\left(\frac{a_i + \frac{\rho}{\sigma_v}\left(y_{2i}-\mathbf{x_{i}^{\prime}} \bm{\beta} + S\sigma_u|U_i|\right)}{\sqrt{1-\rho^2}}\right)p\left(|U_i|\right)d|U_i|

where a_i = \mathbf{Z_{si}^{\prime}} \hat{\bm{\gamma}}. This likelihood can be estimated using five different approaches: Gauss-Kronrod quadrature, adaptive integration over hypercubes (hcubature and pcubature), Gauss-Hermite quadrature, and maximum simulated likelihood. We also use the BHHH estimator to obtain the asymptotic standard errors for the parameter estimators.

sfaselectioncross 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 'sfaselectioncross' 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

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

Note

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

References

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.

Dakpo, K. H., Latruffe, L., Desjeux, Y., Jeanneaux, P., 2022. Modeling heterogeneous technologies in the presence of sample selection: The case of dairy farms and the adoption of agri-environmental schemes in France. Agricultural Economics, 53(3), 422-438.

Greene, W., 2010. A stochastic frontier model with correction for sample selection. Journal of Productivity Analysis. 34, 15–24.

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

Heckman, J., 1976. Discrete, qualitative and limited dependent variables. Ann Econ Soc Meas. 4, 475–492.

Heckman, J., 1979. Sample Selection Bias as a Specification Error. Econometrica. 47, 153–161.

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.

See Also

print for printing sfaselectioncross 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

## Not run: 

## Simulated example

N <- 2000  # sample size
set.seed(12345)
z1 <- rnorm(N)
z2 <- rnorm(N)
v1 <- rnorm(N)
v2 <- rnorm(N)
e1 <- v1
e2 <- 0.7071 * (v1 + v2)
ds <- z1 + z2 + e1
d <- ifelse(ds > 0, 1, 0)
u <- abs(rnorm(N))
x1 <- rnorm(N)
x2 <- rnorm(N)
y <- x1 + x2 + e2 - u
data <- cbind(y = y, x1 = x1, x2 = x2, z1 = z1, z2 = z2, d = d)

## Estimation using quadrature (Gauss-Kronrod)

selecRes1 <- sfaselectioncross(selectionF = d ~ z1 + z2, frontierF = y ~ x1 + x2, 
modelType = 'greene10', method = 'bfgs',
logDepVar = TRUE, data = as.data.frame(data),
S = 1L, udist = 'hnormal', lType = 'kronrod', Nsub = 100, uBound = Inf,
simType = 'halton', Nsim = 300, prime = 2L, burn = 10, antithetics = FALSE,
seed = 12345, itermax = 2000, printInfo = FALSE)

summary(selecRes1)

## Estimation using maximum simulated likelihood

selecRes2 <- sfaselectioncross(selectionF = d ~ z1 + z2, frontierF = y ~ x1 + x2, 
modelType = 'greene10', method = 'bfgs',
logDepVar = TRUE, data = as.data.frame(data),
S = 1L, udist = 'hnormal', lType = 'msl', Nsub = 100, uBound = Inf,
simType = 'halton', Nsim = 300, prime = 2L, burn = 10, antithetics = FALSE,
seed = 12345, itermax = 2000, printInfo = FALSE)

summary(selecRes2)


## End(Not run)

Skewness test for stochastic frontier models

Description

skewnessTest computes skewness test for stochastic frontier models (i.e. objects of class 'sfacross').

Usage

skewnessTest(object, test = "agostino")

Arguments

Value

skewnessTest returns the results of either the D'Agostino's or the Coelli's skewness test.

Note

skewnessTest is currently only available for object of class 'sfacross'.

References

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.

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')
skewnessTest(tl_u_ts)
skewnessTest(tl_u_ts, test = 'coelli')

## End(Not run)

Summary of results for stochastic frontier models

Description

Create and print summary results for stochastic frontier models returned by sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

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

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

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

Arguments

Value

The summary method returns a list of class 'summary.sfacross', 'summary.sfalcmcross', or
'summary.sfaselectioncross' that contains the same elements as an object returned by sfacross, sfalcmcross, or sfaselectioncross with the following additional elements:

See Also

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

sfalcmcross, for the latent class stochastic frontier analysis model fitting function for cross-sectional or pooled data.

sfaselectioncross for sample selection in stochastic frontier model fitting function for cross-sectional or pooled data.

print for printing sfacross object.

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.

skewnessTest for implementing skewness test.

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')
summary(tl_u_ts, grad = TRUE, ci = TRUE)

Data on Swiss railway companies

Description

This dataset is an unbalanced panel of 50 Swiss railway companies over the period 1985-1997.

Format

A data frame with 605 observations on the following 42 variables.

ID

Firm identification.

YEAR

Year identification.

NI

Number of years observed.

STOPS

Number of stops in network.

NETWORK

Network length (in meters).

NARROW_T

Dummy variable for railroads with narrow track.

RACK

Dummy variable for ‘rack rail’ in network.

TUNNEL

Dummy variable for network with tunnels over 300 meters on average.

T

Time indicator, first year = 0.

Q2

Passenger output – passenger km.

Q3

Freight output – ton km.

CT

Total cost (1,000 Swiss franc).

PL

Labor price.

PE

Electricity price.

PK

Capital price.

VIRAGE

1 for railroads with curvy tracks.

LNCT

Log of CT/PE.

LNQ2

Log of Q2.

LNQ3

Log of Q3.

LNNET

Log of NETWORK/1000.

LNPL

Log of PL/PE.

LNPE

Log of PE.

LNPK

Log of PK/PE.

LNSTOP

Log of STOPS.

MLNQ2

Mean of LNQ2.

MLNQ3

Mean of LNQ3.

MLNNET

Mean of LNNET.

MLNPL

Mean of LNPL.

MLNPK

Mean of LNPK.

MLNSTOP

Mean of LNSTOP.

Details

The dataset is extracted from the annual reports of the Swiss Federal Office of Statistics on public transport companies and has been used in Farsi et al. (2005).

Source

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

References

Farsi, M., M. Filippini, and W. Greene. 2005. Efficiency measurement in network industries: Application to the Swiss railway companies. Journal of Regulatory Economics, 28:69–90.

Examples

str(swissrailways)

Data on U.S. electricity generating plants

Description

This dataset contains data on fossil fuel fired steam electric power generation plants in the United States between 1986 and 1996.

Format

A data frame with 791 observations on the following 11 variables.

firm

Plant identification.

year

Year identification.

y

Net-steam electric power generation in megawatt-hours.

regu

Dummy variable which takes a value equal to 1 if the power plant is in a state which enacted legislation or issued a regulatory order to implement retail access during the sample period, and 0 otherwise.

k

Capital stock.

labor

Labor and maintenance.

fuel

Fuel.

wl

Labor price.

wf

Fuel price.

wk

Capital price.

tc

Total cost.

Details

The dataset has been used in Kumbhakar et al. (2014).

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(utility)
summary(utility)

Compute variance-covariance matrix of stochastic frontier models

Description

vcov computes the variance-covariance matrix of the maximum likelihood (ML) coefficients from stochastic frontier models estimated with sfacross, sfalcmcross, or sfaselectioncross.

Usage

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

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

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

Arguments

Details

The variance-covariance matrix is obtained by the inversion of the negative Hessian matrix. Depending on the distribution and the 'hessianType' option, the analytical/numeric Hessian or the bhhh Hessian is evaluated.

The argument extraPar, is currently available only for objects of class 'sfacross' and 'sfaselectioncross'. When 'extraPar = TRUE', the variance-covariance of the additional parameters is obtained using the delta method.

Value

The variance-covariance matrix of the maximum likelihood coefficients is returned.

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

## Using data on Spanish dairy farms
# Cobb Douglas (production function) half normal distribution
cb_s_h <- sfacross(formula = YIT ~ X1 + X2 + X3 + X4, udist = 'hnormal',
data = dairyspain, S = 1, method = 'bfgs')
vcov(cb_s_h)
vcov(cb_s_h, extraPar = TRUE)
 
# Other variance-covariance matrices can be obtained using the sandwich package
 
# Robust variance-covariance matrix
 
requireNamespace('sandwich', quietly = TRUE)
 
sandwich::vcovCL(cb_s_h)
 
# Coefficients and standard errors can be obtained using lmtest package
 
requireNamespace('lmtest', quietly = TRUE)
 
lmtest::coeftest(cb_s_h, vcov. = sandwich::vcovCL)
 
# Clustered standard errors
 
lmtest::coeftest(cb_s_h, vcov. = sandwich::vcovCL, cluster = ~ FARM)
 
# Doubly clustered standard errors
 
lmtest::coeftest(cb_s_h, vcov. = sandwich::vcovCL, cluster = ~ FARM + YEAR)
 
# BHHH standard errors
 
lmtest::coeftest(cb_s_h, vcov. = sandwich::vcovOPG)
 
# Adjusted BHHH standard errors
 
lmtest::coeftest(cb_s_h, vcov. = sandwich::vcovOPG, adjust = TRUE)

## 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, S = 1)
vcov(cb_2c_h)

Data on world production

Description

This dataset provides information on production related variables for eighty-two countries over the period 1960–1987.

Format

A data frame with 2,296 observations on the following 12 variables.

country

Country name.

code

Country identification.

yr

Year identification.

y

GDP in 1987 U.S. dollars.

k

Physical capital stock in 1987 U.S. dollars.

l

Labor (number of individuals in the workforce between the age of 15 and 64).

h

Human capital-adjusted labor.

ly

Log of y.

lk

Log of k.

ll

Log of l.

lh

Log of h.

initStat

Log of the initial capital to labor ratio of each country, lk - ll, measured at the beginning of the sample period.

Details

The dataset is from the World Bank STARS database and has been used in Kumbhakar et al. (2014).

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(worldprod)
summary(worldprod)