semsfa (version 1.1)

efficiencies.semsfa: Prediction of the individual efficiency score

Description

This function calculates and returns efficiency estimates from semiparametric stochastic frontier models estimated with semsfa().

Usage

efficiencies.semsfa(semobj, log.output = TRUE, ...)

Value

An object of class semsfa containing the following additional results:

u

the prediction of the individual efficiency score

efficiencies

point estimate of the efficiency

Arguments

semobj

a stochastic frontier model object returned by semsfa()

log.output

logical. Is the dependent variable logged?

...

further arguments to the summary method are currently ignored

Author

Giancarlo Ferrara and Francesco Vidoli

Details

The estimation of the individual efficiency score for a particular point \((x,y)\) on a production frontier might be obtained from the Jondrow et al. (1982) procedure. Defining: $$ \sigma^2=\sigma_u^2+\sigma_v^2, u_{*}(x) = -\sigma_u^2 \epsilon/\sigma^2, \sigma_{*}^2=\sigma_u^2 \sigma_v^2/\sigma^2 $$ it can be shown that: $$ u|\epsilon ~ N^+(\mu_{*}(x),\sigma_{*}^{2}(x)). $$ We can use this distribution to obtain point previsions of \(u\) trought the mean of the conditional distribution: $$ E(u|\epsilon)=\mu_{*} + \sigma_{*} f(-\mu_{*}/\sigma_{*})/(1-F(\mu_{*}/\sigma_{*})) $$ where \(f\) and \(F\) represent the standard Normal density and cumulative distribution function, respectively; alternative formulas for cost frontier models are easy to get (please see Kumbhakar and Lovell, 2000).

If the response variable is measured in logs, a point estimate of the efficiency is then provided by \(\exp(-u) \in (0,1)\); otherwise, (fitt-u)/fitt where fitt is the estimated output evaluated at the frontier, given the inputs.

References

Jondrow, J., Lovell, C.A.K., Materov, I.S., Schmidt, P., 1982. On the estimation of technical inefficiency in stochastic frontier production models. Journal of Econometrics 19, 233-238.

Kumbhakar, S.C., Lovell, C.A.K., 2000. Stochastic Frontier Analysis. Cambridge University Press, New York.

See Also

semsfa, summary.semsfa, plot.semsfa.

Examples

set.seed(0)
n<-200

#generate data
x<- runif(n, 1, 2)
fy<- 2+30*x-5*x^2
v<- rnorm(n, 0, 1)
u<- abs(rnorm(n,0,2.5))
#production frontier
y <- fy + v - u
dati<-data.frame(y,x)

#first-step: gam, second-step: fan (default)
o<-semsfa(y~s(x),dati,sem.method="gam")

#calculate efficiencies
a<-efficiencies.semsfa(o)