stoned: Convex nonparametric least squares

In Benchmarking: Benchmark and Frontier Analysis Using DEA and SFA

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.

No non-commercial solver at this time of writing is able to solve the NLP formulation required for the multiplicative S toned. Therefore, the NLP is approximated with a QP formulation with some transformation of the objective function. Unfortunately this has not been checked in alle details.

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)

Benchmarking documentation built on April 4, 2025, 5:07 a.m.