Description Usage Arguments Details Value Author(s) References See Also Examples
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).
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, ... )
|
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 |
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.
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 ( |
Tim Coelli and Arne Henningsen
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.
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.
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 )
|
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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