sfa {Benchmarking}

Stochastic frontier estimation

Description

Estimate a stochastic frontier production or cost function using a maximum likelihood method.

Usage

sfa(x, y, beta0 = NULL, lambda0 = 1, resfun = ebeta, 
    TRANSPOSE = FALSE, DEBUG=FALSE,
    control=list(), hessian=2)

sfa.cost(W, Y, COST, beta0 = NULL, lambda0 = 1, resfun = ebeta, 
    TRANSPOSE = FALSE, DEBUG=FALSE,
    control=list(), hessian=2)

te.sfa(object)
teBC.sfa(object)
teMode.sfa(object)
teJ.sfa(object)

te.add.sfa(object, ...)

sigma2u.sfa(object)
sigma2v.sfa(object)
sigma2.sfa(object)

lambda.sfa(object)

Arguments

x

Input as a K x m matrix of observations on m inputs from K firms; (firm x input); MUST be a matrix. No constant for the intercept should be included in x as it is added by default.

y

Output; K times 1 matrix (one output)

Y

Output; K times n matrix for m outputs; only to be used in cost function estimation.

W

Input prices as a K x m matrix.

COST

Cost as a K array for the K firms

beta0

Optional initial parameter values

lambda0

Optional initial ratio of variances

resfun

Function to calculate the residuals, default is a linear model with an intercept. Must be called as resfun(x,y,parm) where parm=c(beta,lambda) or parm=c(beta), and return the residuals as an array of length corresponding to the length of output y.

TRANSPOSE

If TRUE, data is transposed, i.e. input is now m x K matrix

DEBUG

Set to TRUE to get various debugging information written on the console

control

List of control parameters to ucminf

hessian

How the Hessian is delivered, see the ucminf documentation

object

Object of class ‘sfa’ as output from the function sfa

...

Further arguments ...

Details

The optimization is done by the R method ucminf from the package with the same name. The efficiency terms are assumed to be half–normal distributed.

Changing the maximum step length, the trust region, might be important, and this can be done by the option 'control = list(stepmax=0.1)'. The default value is 0.1 and that value is suitable for parameters around 1; for smaller parameters a lower value should be used. Notice that the step length is updated by the optimizing program and thus, must be set for every call of the function sfa if it is to be set.

The generic functions print.sfa, summary.sfa, fitted.sfa, residuals.sfa, logLik.sfa, and coef.sfa all work as expected.

The methods te.sfa, teMode.sfa etc. calculates the efficiency corresponding to different methods

Value

The values returned from sfa is the same as for ucminf, i.e. a list with components plus some especially relevant for sfa:

par

The best set of parameters found c(beta,lambda).

value

The value of minus log-likelihood function corresponding to 'par'.

beta

The parameters for the function

sigma2

The estimate of the total variance

lambda

The estimate of lambda

N

The number of observations

df

The degrees of freedom for the model

residuals

The residuals as a K times 1 matrix/vector, can also be obtained by
residuals(sfa-object)

fitted.values

Fitted values

vcov

The variance-covarians matrix for all estimated parameters incl. lambda

convergence

An integer code. '0' indicates successful convergence. Some of the error codes taken from ucminf are

'1' Stopped by small gradient (grtol).

'2' Stopped by small step (xtol).

'3' Stopped by function evaluation limit (maxeval).

'4' Stopped by zero step from line search

More codes are found in ucminf

message

A character string giving any additional information returned by the optimizer, or 'NULL'.

o

The object returned by ucminf, for further information on this see ucminf

Note

Calculation of technical efficiencies for each unit can be done by the method te.sfa as shown in the examples.

te.sfa(sfaObject), teBC.sfa(sfaObject): Efficiencies estimated by minimizing the mean square error; Eq. (7.21) in Bogetoft and Otto (2011, 219) and Battese and Coelli (1988, 392)

teMode.sfa(sfaObject), te1.sfa(sfaObject): Efficiencies estimates using the conditional mode approach; Bogetoft and Otto (2011, 219), Jondrow et al. (1982, 235).

teJ.sfa(sfaObject), te2.sfa(sfaObject): Efficiencies estimates using the conditional mean approach Jondrow et al. (1982, 235).

te.add.sfa(sfaObject) Efficiency in the additive model, Bogetoft and Otto (2011, 219)

The variance pf the distribution of efficiency can be calculated by sigma2u.sfa(sfaObject), the variance of the random error by sigma2v.sfa(sfaObject), and the total variance (sum of variances of efficiency and random noise) by sigma2.sfa.

The ratio of variances of the efficiency and the random noise can be found from the method lambda.sfa

The generic method summary prints the parameters, standard errors, t-values, and a few more statistics from the optimization.

Author(s)

Peter Bogetoft and Lars Otto [e-mail]

References

Bogetoft and Otto; Benchmarking with DEA, SFA, and R, Springer 2011; chapters 7 and 8.

See Also

See the method ucminf for the possible optimization methods and further options to use in the option control.

The method sfa in the package frontier gives another way to estimate stochastic production functions.

Examples

# Example from the book by Coelli et al.
# d <- read.csv("c:/0work/rpack/front41Data.csv", header = TRUE, sep = ",")
# x <- cbind(log(d$capital), log(d$labour))
# y <- matrix(log(d$output))

n <- 50
x1 <- 1:50 + rnorm(n, 0, 10)
x2 <- 100 + rnorm(n, 0, 10)
x <- cbind(x1, x2)
y <- 0.5 + 1.5*x1 + 2*x2 + rnorm(n, 0, 1) - pmax(0, rnorm(n, 0, 1))
sfa(x,y)
summary(sfa(x,y))


# Estimate efficiency for each unit
o <- sfa(x,y)
eff(o)

te <- te.sfa(o)
teM <- teMode.sfa(o)
teJ <- teJ.sfa(o)
cbind(eff(o),te,Mode=eff(o, type="Mode"),teM,teJ)[1:10,]


sigma2.sfa(o)       # Estimated varians
lambda.sfa(o)       # Estimated lambda