comper {dsfa} R Documentation

## comper

### Description

The comper implements the composed-error distribution in which the \mu, \sigma_V and \sigma_U can depend on additive predictors. Useable with mgcv::gam, the additive predictors are specified via a list of formulae.

### Usage

comper(
s = -1,
distr = "normhnorm",
b = 0.01
)


### Arguments

 link three item list, specifying the link for the \mu, \sigma_V and \sigma_U parameters. See details. s integer; s=-1 for production and s=1 for cost function. distr string; determines the distribution: 'normhnorm', Normal-halfnormal distribution 'normexp', Normal-exponential distribution b positive parameter of the logshift link function.

### Details

Used with gam() to fit distributional stochastic frontier model. The function is called with a list containing three formulae:

1. The first formula specifies the response on the left hand side and the structure of the additive predictor for \mu parameter on the right hand side. Link function is "identity".

2. The second formula is one sided, specifying the additive predictor for the \sigma_V on the right hand side. Link function is "logshift", e.g. \log \{ \sigma_V \} + b .

3. The third formula is one sided, specifying the additive predictor for the \sigma_U on the right hand side. Link function is "logshift", e.g. \log \{ \sigma_U \} + b .

The fitted values and linear predictors for this family will be three column matrices. The first column is the \mu, the second column is the \sigma_V, the third column is \sigma_U. For more details of the distribution see dcomper().

### Value

An object inheriting from class general.family of the mgcv package, which can be used in the mgcv and dsfa package.

### References

• Schmidt R, Kneib T (2023). “Multivariate distributional stochastic frontier models.” Computational Statistics & Data Analysis, 107796.

• Wood SN, Fasiolo M (2017). “A generalized Fellner-Schall method for smoothing parameter optimization with application to Tweedie location, scale and shape models.” Biometrics, 73(4), 1071–1081.

• Aigner D, Lovell CK, Schmidt P (1977). “Formulation and estimation of stochastic frontier production function models.” Journal of econometrics, 6(1), 21–37.

• Kumbhakar SC, Wang H, Horncastle AP (2015). A practitioner's guide to stochastic frontier analysis using Stata. Cambridge University Press.

• Azzalini A (2013). The skew-normal and related families, volume 3. Cambridge University Press.

• Schmidt R, Kneib T (2020). “Analytic expressions for the Cumulative Distribution Function of the Composed Error Term in Stochastic Frontier Analysis with Truncated Normal and Exponential Inefficiencies.” arXiv preprint arXiv:2006.03459.

### Examples

### First example with simulated data
#Set seed, sample size and type of function
set.seed(1337)
N=500 #Sample size
s=-1 #Set to production function

#Generate covariates
x1<-runif(N,-1,1); x2<-runif(N,-1,1); x3<-runif(N,-1,1)
x4<-runif(N,-1,1); x5<-runif(N,-1,1)

#Set parameters of the distribution
mu=2+0.75*x1+0.4*x2+0.6*x2^2+6*log(x3+2)^(1/4) #production function parameter
sigma_v=exp(-1.5+0.75*x4) #noise parameter
sigma_u=exp(-1+sin(2*pi*x5)) #inefficiency parameter

#Simulate responses and create dataset
y<-rcomper(n=N, mu=mu, sigma_v=sigma_v, sigma_u=sigma_u, s=s, distr="normhnorm")
dat<-data.frame(y, x1, x2, x3, x4, x5)

#Write formulae for parameters
mu_formula<-y~x1+x2+I(x2^2)+s(x3, bs="ps")
sigma_v_formula<-~1+x4
sigma_u_formula<-~1+s(x5, bs="ps")

#Fit model
model<-dsfa(formula=list(mu_formula, sigma_v_formula, sigma_u_formula),
data=dat, family=comper(s=s, distr="normhnorm"), optimizer = c("efs"))

#Model summary
summary(model)

#Smooth effects
#Effect of x3 on the predictor of the production function
plot(model, select=1) #Estimated function
lines(x3[order(x3)], 6*log(x3[order(x3)]+2)^(1/4)-
mean(6*log(x3[order(x3)]+2)^(1/4)), col=2) #True effect

#Effect of x5 on the predictor of the inefficiency
plot(model, select=2) #Estimated function
lines(x5[order(x5)], -1+sin(2*pi*x5)[order(x5)]-
mean(-1+sin(2*pi*x5)),col=2) #True effect

### Second example with real data

data("RiceFarms", package = "plm") #load data
RiceFarms[,c("goutput","size","seed", "totlabor", "urea")]<-
log(RiceFarms[,c("goutput","size","seed", "totlabor", "urea")]) #log outputs and inputs
RiceFarms$id<-factor(RiceFarms$id) #id as factor

#Set to production function
s=-1

#Write formulae for parameters
mu_formula<-goutput ~  s(size, bs="ps") + s(seed, bs="ps") + #non-linear effects
s(totlabor, bs="ps") + s(urea, bs="ps") + #non-linear effects
varieties + #factor
s(id, bs="re") #random effect
sigma_v_formula<-~1
sigma_u_formula<-~bimas

#Fit model with normhnorm dstribution
model<-dsfa(formula=list(mu_formula, sigma_v_formula, sigma_u_formula),
data=RiceFarms, family=comper(s=-1, distr="normhnorm"), optimizer = "efs")

#Summary of model
summary(model)

#Plot smooths
plot(model)

### Third example with real data of cost function

data("electricity", package = "sfaR") #load data

electricity$lcof<-log(electricity$cost/electricity$fprice) electricity$lo<-log(electricity$output) electricity$llf<-log(electricity$lprice/electricity$fprice)
electricity$lcf<-log(electricity$cprice/electricity\$fprice)

#Set to cost function
s=1

#Write formulae for parameters
mu_formula<-lcof ~ s(lo, bs="ps") + s(llf, bs="ps") + s(lcf, bs="ps") #non-linear effects
sigma_v_formula<-~1
sigma_u_formula<-~s(lo, bs="ps") + s(lshare, bs="ps") + s(cshare, bs="ps")

#Fit model with normhnorm dstribution
model<-dsfa(formula=list(mu_formula, sigma_v_formula, sigma_u_formula),
data=electricity, family=comper(s=s, distr="normhnorm"),
optimizer = "efs")

#Summary of model
summary(model)

#Plot smooths
plot(model)



[Package dsfa version 2.0.2 Index]