# efficiencies.semsfa: Prediction of the individual efficiency score In semsfa: Semiparametric Estimation of Stochastic Frontier Models

## Description

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

## Usage

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

## 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

## 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:

σ^2=σ_u^2+σ_v^2, u_{*}(x) = -σ_u^2 ε/σ^2, σ_{*}^2=σ_u^2 σ_v^2/σ^2

it can be shown that:

u|ε ~ N^+(μ_{*}(x),σ_{*}^{2}(x)).

We can use this distribution to obtain point previsions of u trought the mean of the conditional distribution:

E(u|ε)=μ_{*} + σ_{*} f(-μ_{*}/σ_{*})/(1-F(μ_{*}/σ_{*}))

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.

## 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

## Author(s)

Giancarlo Ferrara and Francesco Vidoli

## 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.

semsfa, summary.semsfa, plot.semsfa.

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 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) 

### Example output

Loading required package: mgcv