sfa {frontier}R Documentation

Stochastic Frontier Analysis

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

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, 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 plm.data), 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.
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).
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.
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 arne.henningsen@gmail.com

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.htm, 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, and lrtest.frontier for comparing models by LR tests.

Examples

   # 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 )

   # rice producers in the Philippines (panel data)
   data( riceProdPhil )
   riceProdPhil <- plm.data( riceProdPhil, c( "FMERCODE", "YEARDUM" ) )

   # Error Components Frontier (Battese & Coelli 1992)
   rice <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ),
      data = riceProdPhil )
   summary( rice )

   # Technical Efficiency Effects Frontier (Battese & Coelli 1995)
   rice2 <- sfa( log( PROD ) ~ log( AREA ) + log( LABOR ) + log( NPK ) |
      EDYRS + BANRAT - 1, data = riceProdPhil )
   summary( rice2 )

   # 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 )

   ######################################################
   ### using frontier() with the "old" user interface ###
   ######################################################

   # example included in FRONTIER 4.1 (cross-section data)
   front41Data$logOutput  <- log( front41Data$output )
   front41Data$logCapital <- log( front41Data$capital )
   front41Data$logLabour  <- log( front41Data$labour )

   # Cobb-Douglas production frontier
   cobbDouglasFront <- frontier( yName = "logOutput",
      xNames = c( "logCapital", "logLabour" ), data = front41Data )
   all.equal( cobbDouglasFront, cobbDouglas )   # only the call (element 34)
      # and the names of the parameters are slightly different

   # rice producers in the Philippines (panel data)
   riceProdPhil$lPROD  <- log( riceProdPhil$PROD )
   riceProdPhil$lAREA  <- log( riceProdPhil$AREA )
   riceProdPhil$lLABOR <- log( riceProdPhil$LABOR )
   riceProdPhil$lNPK   <- log( riceProdPhil$NPK )

   # Error Components Frontier (Battese & Coelli 1992)
   riceFront <- frontier( yName = "lPROD",
      xNames = c( "lAREA", "lLABOR", "lNPK" ), data = riceProdPhil )
   all.equal( riceFront, rice )   # only the call (element 34)
      # and the names of the parameters are slightly different

   # Technical Efficiency Effects Frontier (Battese & Coelli 1995)
   rice2Front <- frontier( yName = "lPROD",
      xNames = c( "lAREA", "lLABOR", "lNPK" ),
      zNames = c( "EDYRS", "BANRAT" ), data = riceProdPhil )
   all.equal( rice2Front, rice2 )   # only the call (element 34)
      # and the names of the parameters are slightly different

[Package frontier version 0.996-4 Index]