frontier: Stochastic Frontier Analysis
In frontier: Stochastic Frontier Analysis
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
Maximum Likelihood Estimation of
Stochastic Frontier Production and Cost Functions.
Two specifications are available:
the error components specification with time-varying efficiencies
(Battese and Coelli 1992)
and a model specification in which the firm effects are directly
influenced by a number of variables (Battese and Coelli 1995).
This R package uses the Fortran source code of Frontier 4.1
(Coelli 1996).
Usage
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sfa( formula, data = sys.frame( sys.parent() ),
ineffDecrease = TRUE, truncNorm = FALSE,
timeEffect = FALSE, startVal = NULL,
tol = 0.00001, maxit = 1000, muBound = 2, bignum = 1.0E+16,
searchStep = 0.00001, searchTol = 0.001, searchScale = NA,
gridSize = 0.1, gridDouble = TRUE,
restartMax = 10, restartFactor = 0.999, printIter = 0 )
frontier( yName, xNames = NULL, zNames = NULL, data,
zIntercept = FALSE, ... )
## S3 method for class 'frontier'
print( x, digits = NULL, ... )
Arguments
formula
a symbolic description of the model to be estimated;
it can be either a (usual) one-part or a two-part formula
(see section ‘Details’).
data
a (panel) data frame that contains the data;
if data is a usual data.frame,
it is assumed that these are cross-section data;
if data is a panel data frame
(created with pdata.frame),
it is assumed that these are panel data.
ineffDecrease
logical. If TRUE,
inefficiency decreases the endogenous variable
(e.g. for estimating a production function);
if FALSE,
inefficiency increases the endogenous variable
(e.g. for estimating a cost function).
truncNorm
logical. If TRUE,
the inefficiencies are assumed to have a truncated normal distribution
(i.e. parameter mu is added);
if FALSE,
they are assumed to have a half-normal distribution
(only relevant for the ‘Error Components Frontier’).
timeEffect
logical. If FALSE (default),
the efficiency estimates of an ‘Error Components Frontier’
are time invariant;
if TRUE, time is allowed to have an effect on efficiency
(this argument is ignored in case of an
‘Efficiency Effects Frontier’).
startVal
numeric vector. Optional starting values for the ML
estimation.
tol
numeric. Convergence tolerance (proportional).
maxit
numeric. Maximum number of iterations permitted.
muBound
numeric. Bounds on the parameter mu
(see ‘details’ section).
bignum
numeric. Used to set bounds on densities and distributions.
searchStep
numeric. Size of the first step in the Coggin
uni-dimensional search procedure done each iteration
to determine the optimal step length for the next iteration
(see Himmelblau 1972).
searchTol
numeric. Tolerance used in the Coggin
uni-dimensional search procedure done each iteration
to determine the optimal step length for the next iteration
(see Himmelblau 1972).
searchScale
logical or NA. Scaling in the Coggin
uni-dimensional search procedure done each iteration
to determine the optimal step length for the next iteration
(see Himmelblau 1972):
if TRUE, the step length is scaled to the length of the last step;
if FALSE, the step length is not scaled;
if NA, the step length is scaled (to the length of last step)
only if the last step was smaller.
gridSize
numeric. The size of the increment
in the first phase grid search on gamma.
gridDouble
logical. If TRUE,
a second phase grid search on gamma is conducted
around the “best” value obtained in the first phase
with an increment of gridSize/10.
restartMax
integer: maximum number of restarts of the search procedure
when it cannot find a parameter vector
that results in a log-likelihood value
larger than the log-likelihood value of the initial parameters.
restartFactor
numeric scalar: if the search procedure
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value of the initial parameters,
the initial values
(provided by argument startVal or obtained by the grid serach)
are multiplied by this number before the search procedure
is restarted.
printIter
numeric. Print info every printIter iterations;
if this argument is 0, do not print.
yName
string: name of the endogenous variable.
xNames
a vector of strings containing the names of the X variables
(exogenous variables of the production or cost function).
zNames
a vector of strings containing the names of the Z variables
(variables explaining the efficiency level).
zIntercept
logical. If TRUE,
an intercept (with parameter delta_0)
is added to the Z variables
(only relevant for the ‘Efficiency Effects Frontier’).
x
an object of class frontier
(returned by the function frontier).
digits
a non-null value for ‘digits’ specifies
the minimum number of significant digits to be printed in values.
The default, NULL, uses
max(3,getOption("digits")-3).
Non-integer values will be rounded down, and only values greater
than or equal to 1 and no greater than 22 are accepted.
...
additional arguments of frontier
are passed to sfa;
additional arguments of the print method
are currently ignored.
Details
Function frontier is a wrapper function
that calls sfa for the estimation.
The two functions differ only in the user interface;
function frontier has the “old” user interface
and is kept to maintain compatibility with older versions
of the frontier package.
One can use functions sfa and frontier
to calculate the log likelihood value for a given model,
a given data set, and given parameters
by using the argument startVal to specify the parameters
and using the other arguments to specify the model and the data.
The log likelihood value can then be retrieved by
the logLik method
with argument which set to "start".
Setting argument maxit to 0 avoids the
(eventually time-consuming) ML estimation and allows
to retrieve the log likelihood value
with the logLik method
without further arguments.
The frontier function uses the Fortran source code of
Tim Coelli's software FRONTIER 4.1
(http://www.uq.edu.au/economics/cepa/frontier.htm)
and hence, provides the same features as FRONTIER 4.1.
A comprehensive documentation of FRONTIER 4.1 is available
in the file Front41.pdf
that is included in the archive FRONT41-xp1.zip,
which is available at
http://www.uq.edu.au/economics/cepa/frontier.htm.
It is recommended to read this documentation,
because the frontier function is based on the FRONTIER 4.1 software.
If argument formula of sfa is a (usual) one-part formula
(or argument zNames of frontier is NULL),
an ‘Error Components Frontier’ (ECF, see Battese and Coelli 1992)
is estimated.
If argument formula is a two-part formula
(or zNames is not NULL),
an ‘Efficiency Effects Frontier’ (EEF, see Battese and Coelli 1995)
is estimated.
In this case, the first part of the formula
(i.e. the part before the “|” symbol)
is used to explain the endogenous variable directly (X variables),
while the second part of the formula
(i.e. the part after the “|” symbol)
is used to explain the efficiency levels (Z variables).
Generally, there should be no reason for estimating an EEF
without Z variables,
but this can done by setting the second part of argument formula
to 1 (with Z intercept) or - 1 (without Z intercept)
(or by setting argument zNames) to NA).
In case of an Error Components Frontier (ECF)
with the inefficiency terms u following a
truncated normal distribution with mean mu,
argument muBound can be used to restrict mu
to be in the interval +/-muBound * sigma_u,
where sigma_u is the standard deviation of u.
If muBound is infinity, zero, or negative,
no bounds on mu are imposed.
Value
sfa and frontier return a list of class frontier
containing following elements:
modelType
integer.
A ‘1’ denotes an ‘Error Components Frontier’ (ECF);
a ‘2’ denotes an ‘Efficiency Effects Frontier’ (EFF).
ineffDecrease
logical. Argument ineffDecrease (see above).
nn
number of cross-sections.
nt
number of time periods.
nob
number of observations in total.
nb
number of regressor variables (Xs).
truncNorm
logical. Argument truncNorm.
zIntercept
logical. Argument zIntercept.
timeEffect
logical. Argument timeEffect.
printIter
numeric. Argument printIter (see above).
searchScale
numeric. Argument searchScale (see above).
tol
numeric. Argument tol (see above).
searchTol
numeric. Argument searchTol (see above).
bignum
numeric. Argument bignum (see above).
searchStep
numeric. Argument searchStep (see above).
gridDouble
logical. Argument gridDouble (see above).
gridSize
numeric. Argument gridSize (see above).
maxit
numeric. Argument maxit (see above).
muBound
numeric. Argument muBound (see above).
restartMax
numeric. Argument restartMax (see above).
restartFactor
numeric. Argument restartFactor (see above).
nRestart
numeric. Number of restarts of the search procedure
when it cannot find a parameter vector
that results in a log-likelihood value
larger than the log-likelihood value of the initial parameters.
startVal
numeric vector. Argument startVal
(only if specified by user).
call
the matched call.
dataTable
matrix. Data matrix sent to Frontier 4.1.
olsParam
numeric vector. OLS estimates.
olsStdEr
numeric vector. Standard errors of OLS estimates.
olsLogl
numeric. Log likelihood value of OLS estimation.
olsResid
numeric vector. Residuals of the OLS estimation.
olsSkewness
numeric. Skewness of the residuals of the OLS estimation.
olsSkewnessOkay
logical. Indicating if the residuals
of the OLS estimation have the expected skewness.
gridParam
numeric vector. Parameters obtained from the grid search
(if no starting values were specified).
gridLogl
numeric. Log likelihood value of the parameters
obtained from the grid search
(only if no starting values were specified).
startLogl
numeric. Log likelihood value of the starting values
for the parameters
(only if starting values were specified).
mleParam
numeric vector. Parameters obtained from ML estimation.
mleCov
matrix. Covariance matrix of the parameters obtained
from the OLS estimation.
mleLogl
numeric. Log likelihood value of the ML estimation.
nIter
numeric. Number of iterations of the ML estimation.
code
integer indication the reason for determination:
1 = log likelihood values and parameters
of two successive iterations are within the tolerance limits;
5 = cannot find a parameter vector that results
in a log-likelihood value larger than
the log-likelihood value obtained in the previous step;
6 = search failed on gradient step;
10 = maximum number of iterations reached.
nFuncEval
Number of evaluations of the log likelihood function
during the grid search and the iterative ML estimation.
fitted
matrix. Fitted “frontier” values
of the dependent variable:
each row corresponds to a cross-section;
each column corresponds to a time period.
resid
matrix. Residuals:
each row corresponds to a cross-section;
each column corresponds to a time period.
validObs
vector of logical values indicating which observations
of the provided data were used for the estimation,
i.e. do not have values that are not available (NA, NaN)
or infinite (Inf).
Author(s)
Tim Coelli and Arne Henningsen
References
Battese, G.E. and T. Coelli (1992), Frontier production functions,
technical efficiency and panel data: with application to paddy
farmers in India. Journal of Productivity Analysis, 3, 153-169.
Battese, G.E. and T. Coelli (1995), A model for technical inefficiency effects
in a stochastic frontier production function for panel data.
Empirical Economics, 20, 325-332.
Coelli, T. (1996) A Guide to FRONTIER Version 4.1: A Computer
Program for Stochastic Frontier Production and Cost Function
Estimation, CEPA Working Paper 96/08,
http://www.uq.edu.au/economics/cepa/frontier.php,
University of New England.
Himmelblau, D.M. (1972), Applied Non-Linear Programming,
McGraw-Hill, New York.
See Also
frontierQuad for quadratic/translog frontiers,
summary.frontier for creating and printing summary results,
efficiencies.frontier for calculating efficiency estimates,
lrtest.frontier for comparing models by LR tests,
fitted.frontier for obtaining the fitted “frontier” values,
ang residuals.frontier for obtaining the residuals.
Examples
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# example included in FRONTIER 4.1 (cross-section data)
data( front41Data )
# Cobb-Douglas production frontier
cobbDouglas <- sfa( log( output ) ~ log( capital ) + log( labour ),
data = front41Data )
summary( cobbDouglas )
# load data about rice producers in the Philippines (panel data)
data( riceProdPhil )
# Error Components Frontier (Battese & Coelli 1992)
# with observation-specific efficiencies (ignoring the panel structure)
rice <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = riceProdPhil )
summary( rice )
# Error Components Frontier (Battese & Coelli 1992)
# with "true" fixed individual effects and observation-specific efficiencies
riceTrue <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) +
factor( FMERCODE ), data = riceProdPhil )
summary( riceTrue )
# add data set with information about its panel structure
library( "plm" )
ricePanel <- pdata.frame( riceProdPhil, c( "FMERCODE", "YEARDUM" ) )
# Error Components Frontier (Battese & Coelli 1992)
# with time-invariant efficiencies
riceTimeInv <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = ricePanel )
summary( riceTimeInv )
# Error Components Frontier (Battese & Coelli 1992)
# with time-variant efficiencies
riceTimeVar <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = ricePanel, timeEffect = TRUE )
summary( riceTimeVar )
# Technical Efficiency Effects Frontier (Battese & Coelli 1995)
# (efficiency effects model with intercept)
riceZ <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) |
EDYRS + BANRAT, data = riceProdPhil )
summary( riceZ )
# Technical Efficiency Effects Frontier (Battese & Coelli 1995)
# (efficiency effects model without intercept)
riceZ2 <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) |
EDYRS + BANRAT - 1, data = riceProdPhil )
summary( riceZ2 )
# Cost Frontier (with land as quasi-fixed input)
riceProdPhil$cost <- riceProdPhil$LABOR * riceProdPhil$LABORP +
riceProdPhil$NPK * riceProdPhil$NPKP
riceCost <- sfa( log( cost ) ~ log( PROD ) + log( AREA ) + log( LABORP )
+ log( NPKP ), data = riceProdPhil, ineffDecrease = FALSE )
summary( riceCost )
Example output
Loading required package: micEcon
If you have questions, suggestions, or comments regarding one of the 'micEcon' packages, please use a forum or 'tracker' at micEcon's R-Forge site:
https://r-forge.r-project.org/projects/micecon/
Loading required package: lmtest
Loading required package: zoo
Attaching package: 'zoo'
The following objects are masked from 'package:base':
as.Date, as.Date.numeric
Please cite the 'frontier' package as:
Tim Coelli and Arne Henningsen (2013). frontier: Stochastic Frontier Analysis. R package version 1.1. http://CRAN.R-Project.org/package=frontier.
If you have questions, suggestions, or comments regarding the 'frontier' package, please use a forum or 'tracker' at frontier's R-Forge site:
https://r-forge.r-project.org/projects/frontier/
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 7 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.561619 0.202617 2.7718 0.0055742 **
log(capital) 0.281102 0.047643 5.9001 3.632e-09 ***
log(labour) 0.536480 0.045252 11.8555 < 2.2e-16 ***
sigmaSq 0.217000 0.063909 3.3955 0.0006851 ***
gamma 0.797207 0.136424 5.8436 5.109e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -17.02722
cross-sectional data
total number of observations = 60
mean efficiency: 0.7405678
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 9 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.043183 0.252212 -4.1361 3.532e-05 ***
log(AREA) 0.355520 0.060336 5.8923 3.808e-09 ***
log(LABOR) 0.333288 0.062692 5.3163 1.059e-07 ***
log(NPK) 0.271276 0.035238 7.6984 1.378e-14 ***
sigmaSq 0.238634 0.026750 8.9208 < 2.2e-16 ***
gamma 0.885391 0.035545 24.9093 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -86.20268
cross-sectional data
total number of observations = 344
mean efficiency: 0.7229731
Warning message:
In sfa(log(PROD) ~ log(AREA) + log(LABOR) + log(NPK) + factor(FMERCODE), :
the parameter 'gamma' is close to the boundary of the parameter space [0,1]: this can cause convergence problems and can negatively affect the validity and reliability of statistical tests and might be caused by model misspecification
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 36 iterations:
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value obtained in the previous step
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.14901871 0.50175482 0.2970 0.76647
log(AREA) 0.53155134 0.08227644 6.4606 1.043e-10 ***
log(LABOR) 0.19377818 0.11052201 1.7533 0.07955 .
log(NPK) 0.12389081 0.07741476 1.6004 0.10952
factor(FMERCODE)2 0.27513422 0.77815475 0.3536 0.72366
factor(FMERCODE)3 -0.01343728 0.58862288 -0.0228 0.98179
factor(FMERCODE)4 0.20057297 0.67502901 0.2971 0.76637
factor(FMERCODE)5 -0.03117240 0.39710879 -0.0785 0.93743
factor(FMERCODE)6 -0.19563666 0.80169651 -0.2440 0.80721
factor(FMERCODE)7 0.15801987 0.71936788 0.2197 0.82613
factor(FMERCODE)8 0.04052639 0.88469483 0.0458 0.96346
factor(FMERCODE)9 0.04479220 0.63256363 0.0708 0.94355
factor(FMERCODE)10 0.15591061 0.90545593 0.1722 0.86329
factor(FMERCODE)11 -0.30092416 0.24749552 -1.2159 0.22403
factor(FMERCODE)12 0.20205519 0.75543029 0.2675 0.78911
factor(FMERCODE)13 -0.09055144 0.91246418 -0.0992 0.92095
factor(FMERCODE)14 -0.00191789 0.62478311 -0.0031 0.99755
factor(FMERCODE)15 -0.38096390 0.60345085 -0.6313 0.52784
factor(FMERCODE)16 -0.12911985 0.83178627 -0.1552 0.87664
factor(FMERCODE)17 0.37877389 0.18927034 2.0012 0.04537 *
factor(FMERCODE)18 0.28977587 0.47352222 0.6120 0.54057
factor(FMERCODE)19 0.39447872 0.19488167 2.0242 0.04295 *
factor(FMERCODE)20 0.13030699 0.57221457 0.2277 0.81986
factor(FMERCODE)21 0.14694707 0.44069363 0.3334 0.73880
factor(FMERCODE)22 0.06868002 0.65553066 0.1048 0.91656
factor(FMERCODE)23 -0.03685168 0.42198817 -0.0873 0.93041
factor(FMERCODE)24 0.00946378 0.49612205 0.0191 0.98478
factor(FMERCODE)25 0.15369959 0.68591585 0.2241 0.82270
factor(FMERCODE)26 -0.06728899 0.60446613 -0.1113 0.91136
factor(FMERCODE)27 0.04860041 0.85920711 0.0566 0.95489
factor(FMERCODE)28 0.08921273 0.80328445 0.1111 0.91157
factor(FMERCODE)29 -0.01676247 0.43853474 -0.0382 0.96951
factor(FMERCODE)30 -0.48339665 0.24290957 -1.9900 0.04659 *
factor(FMERCODE)31 0.22707161 0.20506536 1.1073 0.26816
factor(FMERCODE)32 0.52550734 0.38819505 1.3537 0.17583
factor(FMERCODE)33 -0.00309571 0.87978492 -0.0035 0.99719
factor(FMERCODE)34 -0.71432603 0.33774672 -2.1150 0.03443 *
factor(FMERCODE)35 0.18284349 0.67311351 0.2716 0.78590
factor(FMERCODE)36 0.00068349 0.36595801 0.0019 0.99851
factor(FMERCODE)37 0.45285689 0.52943159 0.8554 0.39235
factor(FMERCODE)38 0.30217252 0.82054086 0.3683 0.71268
factor(FMERCODE)39 -0.14721505 0.71132349 -0.2070 0.83604
factor(FMERCODE)40 -0.15882656 0.75442031 -0.2105 0.83326
factor(FMERCODE)41 0.15199062 0.47236365 0.3218 0.74763
factor(FMERCODE)42 0.16804383 0.61336860 0.2740 0.78411
factor(FMERCODE)43 0.12419902 0.16661692 0.7454 0.45602
sigmaSq 0.21011941 0.01420696 14.7899 < 2.2e-16 ***
gamma 0.99987606 0.00632933 157.9751 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: 0.3139739
cross-sectional data
total number of observations = 344
mean efficiency: 0.7258831
Loading required package: Formula
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 11 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.832169 0.275249 -3.0233 0.0025 **
log(AREA) 0.453897 0.063801 7.1143 1.125e-12 ***
log(LABOR) 0.288924 0.063639 4.5400 5.625e-06 ***
log(NPK) 0.227544 0.040859 5.5690 2.562e-08 ***
sigmaSq 0.155377 0.024202 6.4201 1.362e-10 ***
gamma 0.464312 0.088023 5.2749 1.328e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -86.43042
panel data
number of cross-sections = 43
number of time periods = 8
total number of observations = 344
thus there are 0 observations not in the panel
mean efficiency: 0.8187968
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 13 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.753919 0.278409 -2.7080 0.0067700 **
log(AREA) 0.474918 0.065422 7.2593 3.891e-13 ***
log(LABOR) 0.300096 0.063940 4.6934 2.687e-06 ***
log(NPK) 0.199462 0.042633 4.6786 2.888e-06 ***
sigmaSq 0.129956 0.021767 5.9703 2.368e-09 ***
gamma 0.369635 0.107268 3.4459 0.0005691 ***
time 0.058910 0.031049 1.8973 0.0577859 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -84.55036
panel data
number of cross-sections = 43
number of time periods = 8
total number of observations = 344
thus there are 0 observations not in the panel
mean efficiency of each year
1 2 3 4 5 6 7 8
0.7848440 0.7950311 0.8048370 0.8142661 0.8233236 0.8320156 0.8403494 0.8483325
mean efficiency: 0.8178749
Efficiency Effects Frontier (see Battese & Coelli 1995)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 47 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.051760 0.252990 -4.1573 3.220e-05 ***
log(AREA) 0.379767 0.059976 6.3320 2.421e-10 ***
log(LABOR) 0.321029 0.061125 5.2520 1.505e-07 ***
log(NPK) 0.263797 0.034458 7.6555 1.925e-14 ***
Z_(Intercept) -2.746459 8.499187 -0.3231 0.7466
Z_EDYRS -0.028610 0.215997 -0.1325 0.8946
Z_BANRAT -3.635528 8.096569 -0.4490 0.6534
sigmaSq 1.666588 3.695869 0.4509 0.6520
gamma 0.978799 0.045203 21.6536 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -77.31363
cross-sectional data
total number of observations = 344
mean efficiency: 0.784927
Efficiency Effects Frontier (see Battese & Coelli 1995)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 22 iterations:
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value obtained in the previous step
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.008578 0.246740 -4.0876 4.358e-05 ***
log(AREA) 0.384099 0.059539 6.4512 1.109e-10 ***
log(LABOR) 0.318982 0.061289 5.2045 1.945e-07 ***
log(NPK) 0.260355 0.034417 7.5647 3.887e-14 ***
Z_EDYRS -0.058074 0.087218 -0.6658 0.50551
Z_BANRAT -1.405402 0.735653 -1.9104 0.05608 .
sigmaSq 0.597438 0.296232 2.0168 0.04372 *
gamma 0.944928 0.028488 33.1690 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -77.84954
cross-sectional data
total number of observations = 344
mean efficiency: 0.7707586
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency increases the endogenous variable (as in a cost function)
The dependent variable is logged
Iterative ML estimation terminated after 11 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 5.870429 0.201403 29.1477 < 2.2e-16 ***
log(PROD) 0.461678 0.034405 13.4188 < 2.2e-16 ***
log(AREA) 0.506498 0.035118 14.4226 < 2.2e-16 ***
log(LABORP) 0.413795 0.032272 12.8222 < 2.2e-16 ***
log(NPKP) 0.070512 0.054245 1.2999 0.193642
sigmaSq 0.070379 0.014346 4.9059 9.301e-07 ***
gamma 0.530293 0.187373 2.8301 0.004653 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: 39.93057
cross-sectional data
total number of observations = 344
mean efficiency: 0.8631699
frontier documentation built on April 19, 2020, 3:54 p.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
Maximum Likelihood Estimation of Stochastic Frontier Production and Cost Functions. Two specifications are available: the error components specification with time-varying efficiencies (Battese and Coelli 1992) and a model specification in which the firm effects are directly influenced by a number of variables (Battese and Coelli 1995). This R package uses the Fortran source code of Frontier 4.1 (Coelli 1996).
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 | sfa( formula, data = sys.frame( sys.parent() ),
ineffDecrease = TRUE, truncNorm = FALSE,
timeEffect = FALSE, startVal = NULL,
tol = 0.00001, maxit = 1000, muBound = 2, bignum = 1.0E+16,
searchStep = 0.00001, searchTol = 0.001, searchScale = NA,
gridSize = 0.1, gridDouble = TRUE,
restartMax = 10, restartFactor = 0.999, printIter = 0 )
frontier( yName, xNames = NULL, zNames = NULL, data,
zIntercept = FALSE, ... )
## S3 method for class 'frontier'
print( x, digits = NULL, ... )
|
Arguments
formula |
a symbolic description of the model to be estimated; it can be either a (usual) one-part or a two-part formula (see section ‘Details’). |
data |
a (panel) data frame that contains the data;
if |
ineffDecrease |
logical. If |
truncNorm |
logical. If |
timeEffect |
logical. If |
startVal |
numeric vector. Optional starting values for the ML estimation. |
tol |
numeric. Convergence tolerance (proportional). |
maxit |
numeric. Maximum number of iterations permitted. |
muBound |
numeric. Bounds on the parameter mu (see ‘details’ section). |
bignum |
numeric. Used to set bounds on densities and distributions. |
searchStep |
numeric. Size of the first step in the Coggin uni-dimensional search procedure done each iteration to determine the optimal step length for the next iteration (see Himmelblau 1972). |
searchTol |
numeric. Tolerance used in the Coggin uni-dimensional search procedure done each iteration to determine the optimal step length for the next iteration (see Himmelblau 1972). |
searchScale |
logical or |
gridSize |
numeric. The size of the increment in the first phase grid search on gamma. |
gridDouble |
logical. If |
restartMax |
integer: maximum number of restarts of the search procedure when it cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value of the initial parameters. |
restartFactor |
numeric scalar: if the search procedure
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value of the initial parameters,
the initial values
(provided by argument |
printIter |
numeric. Print info every |
yName |
string: name of the endogenous variable. |
xNames |
a vector of strings containing the names of the X variables (exogenous variables of the production or cost function). |
zNames |
a vector of strings containing the names of the Z variables (variables explaining the efficiency level). |
zIntercept |
logical. If |
x |
an object of class |
digits |
a non-null value for ‘digits’ specifies
the minimum number of significant digits to be printed in values.
The default, |
... |
additional arguments of |
Details
Function frontier is a wrapper function
that calls sfa for the estimation.
The two functions differ only in the user interface;
function frontier has the “old” user interface
and is kept to maintain compatibility with older versions
of the frontier package.
One can use functions sfa and frontier
to calculate the log likelihood value for a given model,
a given data set, and given parameters
by using the argument startVal to specify the parameters
and using the other arguments to specify the model and the data.
The log likelihood value can then be retrieved by
the logLik method
with argument which set to "start".
Setting argument maxit to 0 avoids the
(eventually time-consuming) ML estimation and allows
to retrieve the log likelihood value
with the logLik method
without further arguments.
The frontier function uses the Fortran source code of
Tim Coelli's software FRONTIER 4.1
(http://www.uq.edu.au/economics/cepa/frontier.htm)
and hence, provides the same features as FRONTIER 4.1.
A comprehensive documentation of FRONTIER 4.1 is available
in the file Front41.pdf
that is included in the archive FRONT41-xp1.zip,
which is available at
http://www.uq.edu.au/economics/cepa/frontier.htm.
It is recommended to read this documentation,
because the frontier function is based on the FRONTIER 4.1 software.
If argument formula of sfa is a (usual) one-part formula
(or argument zNames of frontier is NULL),
an ‘Error Components Frontier’ (ECF, see Battese and Coelli 1992)
is estimated.
If argument formula is a two-part formula
(or zNames is not NULL),
an ‘Efficiency Effects Frontier’ (EEF, see Battese and Coelli 1995)
is estimated.
In this case, the first part of the formula
(i.e. the part before the “|” symbol)
is used to explain the endogenous variable directly (X variables),
while the second part of the formula
(i.e. the part after the “|” symbol)
is used to explain the efficiency levels (Z variables).
Generally, there should be no reason for estimating an EEF
without Z variables,
but this can done by setting the second part of argument formula
to 1 (with Z intercept) or - 1 (without Z intercept)
(or by setting argument zNames) to NA).
In case of an Error Components Frontier (ECF)
with the inefficiency terms u following a
truncated normal distribution with mean mu,
argument muBound can be used to restrict mu
to be in the interval +/-muBound * sigma_u,
where sigma_u is the standard deviation of u.
If muBound is infinity, zero, or negative,
no bounds on mu are imposed.
Value
sfa and frontier return a list of class frontier
containing following elements:
modelType |
integer. A ‘1’ denotes an ‘Error Components Frontier’ (ECF); a ‘2’ denotes an ‘Efficiency Effects Frontier’ (EFF). |
ineffDecrease |
logical. Argument |
nn |
number of cross-sections. |
nt |
number of time periods. |
nob |
number of observations in total. |
nb |
number of regressor variables (Xs). |
truncNorm |
logical. Argument |
zIntercept |
logical. Argument |
timeEffect |
logical. Argument |
printIter |
numeric. Argument |
searchScale |
numeric. Argument |
tol |
numeric. Argument |
searchTol |
numeric. Argument |
bignum |
numeric. Argument |
searchStep |
numeric. Argument |
gridDouble |
logical. Argument |
gridSize |
numeric. Argument |
maxit |
numeric. Argument |
muBound |
numeric. Argument |
restartMax |
numeric. Argument |
restartFactor |
numeric. Argument |
nRestart |
numeric. Number of restarts of the search procedure when it cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value of the initial parameters. |
startVal |
numeric vector. Argument |
call |
the matched call. |
dataTable |
matrix. Data matrix sent to Frontier 4.1. |
olsParam |
numeric vector. OLS estimates. |
olsStdEr |
numeric vector. Standard errors of OLS estimates. |
olsLogl |
numeric. Log likelihood value of OLS estimation. |
olsResid |
numeric vector. Residuals of the OLS estimation. |
olsSkewness |
numeric. Skewness of the residuals of the OLS estimation. |
olsSkewnessOkay |
logical. Indicating if the residuals of the OLS estimation have the expected skewness. |
gridParam |
numeric vector. Parameters obtained from the grid search (if no starting values were specified). |
gridLogl |
numeric. Log likelihood value of the parameters obtained from the grid search (only if no starting values were specified). |
startLogl |
numeric. Log likelihood value of the starting values for the parameters (only if starting values were specified). |
mleParam |
numeric vector. Parameters obtained from ML estimation. |
mleCov |
matrix. Covariance matrix of the parameters obtained from the OLS estimation. |
mleLogl |
numeric. Log likelihood value of the ML estimation. |
nIter |
numeric. Number of iterations of the ML estimation. |
code |
integer indication the reason for determination:
|
nFuncEval |
Number of evaluations of the log likelihood function during the grid search and the iterative ML estimation. |
fitted |
matrix. Fitted “frontier” values of the dependent variable: each row corresponds to a cross-section; each column corresponds to a time period. |
resid |
matrix. Residuals: each row corresponds to a cross-section; each column corresponds to a time period. |
validObs |
vector of logical values indicating which observations
of the provided data were used for the estimation,
i.e. do not have values that are not available ( |
Author(s)
Tim Coelli and Arne Henningsen
References
Battese, G.E. and T. Coelli (1992), Frontier production functions, technical efficiency and panel data: with application to paddy farmers in India. Journal of Productivity Analysis, 3, 153-169.
Battese, G.E. and T. Coelli (1995), A model for technical inefficiency effects in a stochastic frontier production function for panel data. Empirical Economics, 20, 325-332.
Coelli, T. (1996) A Guide to FRONTIER Version 4.1: A Computer Program for Stochastic Frontier Production and Cost Function Estimation, CEPA Working Paper 96/08, http://www.uq.edu.au/economics/cepa/frontier.php, University of New England.
Himmelblau, D.M. (1972), Applied Non-Linear Programming, McGraw-Hill, New York.
See Also
frontierQuad for quadratic/translog frontiers,
summary.frontier for creating and printing summary results,
efficiencies.frontier for calculating efficiency estimates,
lrtest.frontier for comparing models by LR tests,
fitted.frontier for obtaining the fitted “frontier” values,
ang residuals.frontier for obtaining the residuals.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | # example included in FRONTIER 4.1 (cross-section data)
data( front41Data )
# Cobb-Douglas production frontier
cobbDouglas <- sfa( log( output ) ~ log( capital ) + log( labour ),
data = front41Data )
summary( cobbDouglas )
# load data about rice producers in the Philippines (panel data)
data( riceProdPhil )
# Error Components Frontier (Battese & Coelli 1992)
# with observation-specific efficiencies (ignoring the panel structure)
rice <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = riceProdPhil )
summary( rice )
# Error Components Frontier (Battese & Coelli 1992)
# with "true" fixed individual effects and observation-specific efficiencies
riceTrue <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) +
factor( FMERCODE ), data = riceProdPhil )
summary( riceTrue )
# add data set with information about its panel structure
library( "plm" )
ricePanel <- pdata.frame( riceProdPhil, c( "FMERCODE", "YEARDUM" ) )
# Error Components Frontier (Battese & Coelli 1992)
# with time-invariant efficiencies
riceTimeInv <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = ricePanel )
summary( riceTimeInv )
# Error Components Frontier (Battese & Coelli 1992)
# with time-variant efficiencies
riceTimeVar <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
data = ricePanel, timeEffect = TRUE )
summary( riceTimeVar )
# Technical Efficiency Effects Frontier (Battese & Coelli 1995)
# (efficiency effects model with intercept)
riceZ <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) |
EDYRS + BANRAT, data = riceProdPhil )
summary( riceZ )
# Technical Efficiency Effects Frontier (Battese & Coelli 1995)
# (efficiency effects model without intercept)
riceZ2 <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) |
EDYRS + BANRAT - 1, data = riceProdPhil )
summary( riceZ2 )
# Cost Frontier (with land as quasi-fixed input)
riceProdPhil$cost <- riceProdPhil$LABOR * riceProdPhil$LABORP +
riceProdPhil$NPK * riceProdPhil$NPKP
riceCost <- sfa( log( cost ) ~ log( PROD ) + log( AREA ) + log( LABORP )
+ log( NPKP ), data = riceProdPhil, ineffDecrease = FALSE )
summary( riceCost )
|
Example output
Loading required package: micEcon
If you have questions, suggestions, or comments regarding one of the 'micEcon' packages, please use a forum or 'tracker' at micEcon's R-Forge site:
https://r-forge.r-project.org/projects/micecon/
Loading required package: lmtest
Loading required package: zoo
Attaching package: 'zoo'
The following objects are masked from 'package:base':
as.Date, as.Date.numeric
Please cite the 'frontier' package as:
Tim Coelli and Arne Henningsen (2013). frontier: Stochastic Frontier Analysis. R package version 1.1. http://CRAN.R-Project.org/package=frontier.
If you have questions, suggestions, or comments regarding the 'frontier' package, please use a forum or 'tracker' at frontier's R-Forge site:
https://r-forge.r-project.org/projects/frontier/
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 7 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.561619 0.202617 2.7718 0.0055742 **
log(capital) 0.281102 0.047643 5.9001 3.632e-09 ***
log(labour) 0.536480 0.045252 11.8555 < 2.2e-16 ***
sigmaSq 0.217000 0.063909 3.3955 0.0006851 ***
gamma 0.797207 0.136424 5.8436 5.109e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -17.02722
cross-sectional data
total number of observations = 60
mean efficiency: 0.7405678
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 9 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.043183 0.252212 -4.1361 3.532e-05 ***
log(AREA) 0.355520 0.060336 5.8923 3.808e-09 ***
log(LABOR) 0.333288 0.062692 5.3163 1.059e-07 ***
log(NPK) 0.271276 0.035238 7.6984 1.378e-14 ***
sigmaSq 0.238634 0.026750 8.9208 < 2.2e-16 ***
gamma 0.885391 0.035545 24.9093 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -86.20268
cross-sectional data
total number of observations = 344
mean efficiency: 0.7229731
Warning message:
In sfa(log(PROD) ~ log(AREA) + log(LABOR) + log(NPK) + factor(FMERCODE), :
the parameter 'gamma' is close to the boundary of the parameter space [0,1]: this can cause convergence problems and can negatively affect the validity and reliability of statistical tests and might be caused by model misspecification
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 36 iterations:
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value obtained in the previous step
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 0.14901871 0.50175482 0.2970 0.76647
log(AREA) 0.53155134 0.08227644 6.4606 1.043e-10 ***
log(LABOR) 0.19377818 0.11052201 1.7533 0.07955 .
log(NPK) 0.12389081 0.07741476 1.6004 0.10952
factor(FMERCODE)2 0.27513422 0.77815475 0.3536 0.72366
factor(FMERCODE)3 -0.01343728 0.58862288 -0.0228 0.98179
factor(FMERCODE)4 0.20057297 0.67502901 0.2971 0.76637
factor(FMERCODE)5 -0.03117240 0.39710879 -0.0785 0.93743
factor(FMERCODE)6 -0.19563666 0.80169651 -0.2440 0.80721
factor(FMERCODE)7 0.15801987 0.71936788 0.2197 0.82613
factor(FMERCODE)8 0.04052639 0.88469483 0.0458 0.96346
factor(FMERCODE)9 0.04479220 0.63256363 0.0708 0.94355
factor(FMERCODE)10 0.15591061 0.90545593 0.1722 0.86329
factor(FMERCODE)11 -0.30092416 0.24749552 -1.2159 0.22403
factor(FMERCODE)12 0.20205519 0.75543029 0.2675 0.78911
factor(FMERCODE)13 -0.09055144 0.91246418 -0.0992 0.92095
factor(FMERCODE)14 -0.00191789 0.62478311 -0.0031 0.99755
factor(FMERCODE)15 -0.38096390 0.60345085 -0.6313 0.52784
factor(FMERCODE)16 -0.12911985 0.83178627 -0.1552 0.87664
factor(FMERCODE)17 0.37877389 0.18927034 2.0012 0.04537 *
factor(FMERCODE)18 0.28977587 0.47352222 0.6120 0.54057
factor(FMERCODE)19 0.39447872 0.19488167 2.0242 0.04295 *
factor(FMERCODE)20 0.13030699 0.57221457 0.2277 0.81986
factor(FMERCODE)21 0.14694707 0.44069363 0.3334 0.73880
factor(FMERCODE)22 0.06868002 0.65553066 0.1048 0.91656
factor(FMERCODE)23 -0.03685168 0.42198817 -0.0873 0.93041
factor(FMERCODE)24 0.00946378 0.49612205 0.0191 0.98478
factor(FMERCODE)25 0.15369959 0.68591585 0.2241 0.82270
factor(FMERCODE)26 -0.06728899 0.60446613 -0.1113 0.91136
factor(FMERCODE)27 0.04860041 0.85920711 0.0566 0.95489
factor(FMERCODE)28 0.08921273 0.80328445 0.1111 0.91157
factor(FMERCODE)29 -0.01676247 0.43853474 -0.0382 0.96951
factor(FMERCODE)30 -0.48339665 0.24290957 -1.9900 0.04659 *
factor(FMERCODE)31 0.22707161 0.20506536 1.1073 0.26816
factor(FMERCODE)32 0.52550734 0.38819505 1.3537 0.17583
factor(FMERCODE)33 -0.00309571 0.87978492 -0.0035 0.99719
factor(FMERCODE)34 -0.71432603 0.33774672 -2.1150 0.03443 *
factor(FMERCODE)35 0.18284349 0.67311351 0.2716 0.78590
factor(FMERCODE)36 0.00068349 0.36595801 0.0019 0.99851
factor(FMERCODE)37 0.45285689 0.52943159 0.8554 0.39235
factor(FMERCODE)38 0.30217252 0.82054086 0.3683 0.71268
factor(FMERCODE)39 -0.14721505 0.71132349 -0.2070 0.83604
factor(FMERCODE)40 -0.15882656 0.75442031 -0.2105 0.83326
factor(FMERCODE)41 0.15199062 0.47236365 0.3218 0.74763
factor(FMERCODE)42 0.16804383 0.61336860 0.2740 0.78411
factor(FMERCODE)43 0.12419902 0.16661692 0.7454 0.45602
sigmaSq 0.21011941 0.01420696 14.7899 < 2.2e-16 ***
gamma 0.99987606 0.00632933 157.9751 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: 0.3139739
cross-sectional data
total number of observations = 344
mean efficiency: 0.7258831
Loading required package: Formula
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 11 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.832169 0.275249 -3.0233 0.0025 **
log(AREA) 0.453897 0.063801 7.1143 1.125e-12 ***
log(LABOR) 0.288924 0.063639 4.5400 5.625e-06 ***
log(NPK) 0.227544 0.040859 5.5690 2.562e-08 ***
sigmaSq 0.155377 0.024202 6.4201 1.362e-10 ***
gamma 0.464312 0.088023 5.2749 1.328e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -86.43042
panel data
number of cross-sections = 43
number of time periods = 8
total number of observations = 344
thus there are 0 observations not in the panel
mean efficiency: 0.8187968
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 13 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -0.753919 0.278409 -2.7080 0.0067700 **
log(AREA) 0.474918 0.065422 7.2593 3.891e-13 ***
log(LABOR) 0.300096 0.063940 4.6934 2.687e-06 ***
log(NPK) 0.199462 0.042633 4.6786 2.888e-06 ***
sigmaSq 0.129956 0.021767 5.9703 2.368e-09 ***
gamma 0.369635 0.107268 3.4459 0.0005691 ***
time 0.058910 0.031049 1.8973 0.0577859 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -84.55036
panel data
number of cross-sections = 43
number of time periods = 8
total number of observations = 344
thus there are 0 observations not in the panel
mean efficiency of each year
1 2 3 4 5 6 7 8
0.7848440 0.7950311 0.8048370 0.8142661 0.8233236 0.8320156 0.8403494 0.8483325
mean efficiency: 0.8178749
Efficiency Effects Frontier (see Battese & Coelli 1995)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 47 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.051760 0.252990 -4.1573 3.220e-05 ***
log(AREA) 0.379767 0.059976 6.3320 2.421e-10 ***
log(LABOR) 0.321029 0.061125 5.2520 1.505e-07 ***
log(NPK) 0.263797 0.034458 7.6555 1.925e-14 ***
Z_(Intercept) -2.746459 8.499187 -0.3231 0.7466
Z_EDYRS -0.028610 0.215997 -0.1325 0.8946
Z_BANRAT -3.635528 8.096569 -0.4490 0.6534
sigmaSq 1.666588 3.695869 0.4509 0.6520
gamma 0.978799 0.045203 21.6536 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -77.31363
cross-sectional data
total number of observations = 344
mean efficiency: 0.784927
Efficiency Effects Frontier (see Battese & Coelli 1995)
Inefficiency decreases the endogenous variable (as in a production function)
The dependent variable is logged
Iterative ML estimation terminated after 22 iterations:
cannot find a parameter vector that results in a log-likelihood value
larger than the log-likelihood value obtained in the previous step
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.008578 0.246740 -4.0876 4.358e-05 ***
log(AREA) 0.384099 0.059539 6.4512 1.109e-10 ***
log(LABOR) 0.318982 0.061289 5.2045 1.945e-07 ***
log(NPK) 0.260355 0.034417 7.5647 3.887e-14 ***
Z_EDYRS -0.058074 0.087218 -0.6658 0.50551
Z_BANRAT -1.405402 0.735653 -1.9104 0.05608 .
sigmaSq 0.597438 0.296232 2.0168 0.04372 *
gamma 0.944928 0.028488 33.1690 < 2.2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: -77.84954
cross-sectional data
total number of observations = 344
mean efficiency: 0.7707586
Error Components Frontier (see Battese & Coelli 1992)
Inefficiency increases the endogenous variable (as in a cost function)
The dependent variable is logged
Iterative ML estimation terminated after 11 iterations:
log likelihood values and parameters of two successive iterations
are within the tolerance limit
final maximum likelihood estimates
Estimate Std. Error z value Pr(>|z|)
(Intercept) 5.870429 0.201403 29.1477 < 2.2e-16 ***
log(PROD) 0.461678 0.034405 13.4188 < 2.2e-16 ***
log(AREA) 0.506498 0.035118 14.4226 < 2.2e-16 ***
log(LABORP) 0.413795 0.032272 12.8222 < 2.2e-16 ***
log(NPKP) 0.070512 0.054245 1.2999 0.193642
sigmaSq 0.070379 0.014346 4.9059 9.301e-07 ***
gamma 0.530293 0.187373 2.8301 0.004653 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
log likelihood value: 39.93057
cross-sectional data
total number of observations = 344
mean efficiency: 0.8631699
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