stoned {Benchmarking}

Convex nonparametric least squares

Description

Convex nonparametric least squares here for convex (Cost) function function or concave (Production) function with multiplicative or additive error term. the StoNED estimator combines the axiomatic and non-parametric frontier (the DEA aspect) with a stochastic noise term (the SFA aspect)

Usage

stoned(X, Y, RTS = "vrs", COST = 0, MULT = 0, METHOD = "MM")

Arguments

X

Inputs (right hand side) of firms to be evaluated, a K x m matrix of observations of K firms with m inputs (firm x input).

Y

Output or cost (left hand side) of firms to be evaluated, a K x 1 matrix of observations of K firms with 1 output or cost (firm x input).

RTS

RTS determines returns to scale assumption: RTS="vrs", "drs", "crs" and "irs" are possible for constant or variable returns to scale; see dea for a verbal description and numbering scheme.

COST

COST specifies whether a cost function needs is estimated (COST=1) or a production function (COST=0).

MULT

MULT determines if multiplicative (MULT=1) or additive (MULT=0) model is estimated.

METHOD

METHOD specifies the way efficiency is estimated: MM for Method of Moments and PSL for pseudo likelihood estimation.

Details

Convex nonparametric least squares here for convex (cost) function with multiplicative error term: Y=b*X*exp(e) or additive error term: Y=b*X + e.

Value

The results are returned in a list with the components:

residualNorm

Norm of residual

solutionNorm

Norm of solution

error

Is there an error in the solution?

coef

beta_matrix, estimated coefficients as a Kxm matrix; if there is an intercept the first column is the intercept, and the matrix is Kx(1+m)

residuals

Residuals

fit

Fitted values

eff

Efficinecy score

front

Points on the frontier

sigma_u

sigma_u

Note

Convex nonparametric least squares here for convex (Cost) function with multiplicative error term: Y=b*X*exp(e) or additive error term: Y=b*X + e.

The intercept is absent for the constant returns to scale assumption; all other technology assumptions do have an intercept.

Note that the method stoned is a rather slow method and probably only works in a reasonable time for less than 3-400 units.

Author(s)

Stefan Seifert [e-mail] and Lars Otto [e-mail]

References

Kuosmanen and Kortelainen, "Stochastic non-smooth envelopment of data: semi-parametric frontier estimation subject to shape constraints", Journal of Productivity Analysis 2012

Examples

#### Example: Single Input Production Function
n=10

x1 <- runif(n,10,20)
v <- rnorm(n,0,0.01)
u <- abs(rnorm(n,0,0.04))

y <- (x1^0.8)*exp(-u)*exp(v)

sol_MM <- stoned(x1, y)
sol_PSL <- stoned(x1, y, METHOD="PSL")

plot(x1,y)
curve(x^0.8, add=TRUE)
points(x1,sol_MM$front, col="red")
points(x1,sol_PSL$front, col="blue", pch=16, cex=.6)