# Stochastic Frontier Analysis 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). ```r 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](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](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. ## Returns `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`). ## 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](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. ## Author(s) Tim Coelli and Arne Henningsen ## Examples ```r # 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 ) ```

frontier function